Funnel plots

Smith and Eggar (1997) in a letter to The Lancet, use funnel plots (shown in Figure 15.2) to illustrate publication bias and to link this with outcomes in several large trials conducted subsequent to the trials within those plots. 

In the upper part of Figure 15.2 we see a funnel plot of trials evaluating the effect of intravenous magnesium in the treatment of myocardial infarction. Note the absence of small trials with odds ratios greater than one (which would indicate a lack of benefit for intravenous magnesium) this... 

Figure 15.2 Funnel plot for meta-analysis of trials of magnesium (upper diagram) and streptokinase (lower diagram) in acute myocardial infarction. (Smith and Eggar (1997). Reproduced with kind permission from The Lancet.)...

In contrast, the lower part of Figure 15.2 shows a funnel plot for trials evaluating streptokinase and the meta-analysis, which combined these trials. The pattern of the individual trial results supports the absence of publication bias and the two large trials shown in the plot, ISIS-2 and GISS-1 show treatment effects entirely in line with the earlier meta-analysis. 

A funnel plot is when studies are plotted according to their size and effect. Studies should be spread out symmetrically around the average effect. If smaller negative studies are not published the plot is asymmetrical. 

Fig. 6.3. To ensure the accuracy of a nonequilibrium work free energy calculation, the switching paths should go down the funnel. The important phase space regions for the intermediate states along the ideal funnel paths are illustrated in this plot, for the case where r0 and / are partially overlapped. Two funnel paths need to be constructed to transfer the systems from both 0 and 1 to a common intermediate M where rm is inside the r0 and J overlap region. The construction of such paths is discussed in Sect. 6.6...

Figure 9 Plot showing derived function used in calculating ratholing potential in funnel flow bins.

Small-scale field trials with fully stabilised micro-encapsulated formulations were carried out in Egypt during 1980 on two cotton pests - pink bollworm, Peotinophora gossypiella, and the Egyptian cotton leafworm, Spodoptera littoralis. Plots of 10 n x 10m were sprayed with the formulation and there were three replicates of each treatment. A plastic funnel trap (3) baited with synthetic pheromone was positioned at the centre of each plot. The effectiveness and persistence of communication disruption were measured by the reduction in catches of male moths in the traps in the treated plots relative to the catches in similar traps in untreated, control plots. 

FIGURE 13 Plot of derived function G(ikt) u.sed to calculate critical rathole diameter for funnel flow bins. 

The result of such column inlet and outlet design is shown in Fig. 3.9. The concentration profile is plotted against dimensionless eluted volume for the cylindrical and exponential funnel. As can be seen, the exponential funnel having the same volume exhibits a much steeper peak. 

If the upper and lower bonds diverge and the pattern of the residuals in the plot is funnel-like, (Fig. 6.11a), it is an indication that the error is different in different parts of the experimental domain. In such cases, the assumption of a constant error variance is violated. To overcome this difficulty and to obtain a fairly constant error distribution, the metric of the response variable should be changed through some transformation, and the modell refitted to the transformed response. How this can be done is discussed in Chapter 12. 

Indications of this can be obtained by plotting tbe residuals in different ways, to make sure that th do not show any abnormal pattern. Sometimes it is found that the plot of the residuals against the predicted response has a funnel-like pattern, which shows that the size of tbe residuals are dependent on the magnitude of tbe responses. A situation when this is often encountered is when integrated chromatographic peaks are used to determine concentrations in tbe sample. When such patterns are observed it is a clear indication that the assumption of a constant variance is violated. This obstacle can sometimes be removed by a mathematical transformation of tbe response variable y... 

J. D. Bryngelson, J. N. Onuchic, N. D. Socci et al. Funnels, pathways, and the energy landscape of protein-folding - a synthesis. Proteins, 21 (1995), 167 D. K. Klimov and D. Thirumalai. Criterion that determines the foldability of proteins. Physical Review Letters, 76 (1996), 4070 H. S. Chan and K. A. Dill. Protein folding in the landscape perspective chevron plots and non-arrhenius kinetics. Proteins, 30 (1998), 2. 

Fig. 6.13 Plot of carbon and nitrogen isotopes in pottery from Mesolithic (Tybrind Vig and Ringkloster) and Neolithic sites (Funnel Beaker). A distinct separation of these two groups is seen reflecting the more terrestrial diet of the Neolithic farmers. This pattern is also seen in the carbon and nitrogen isotope ratios in human bone collagen from the Mesolithic and Neolithic in this region...

The simplest method is to mix solvent and feed liquors containing varying quantities of solute in a separatory funnel, and analyze each phase for solute after settling. Where feed and solvent are essentially immiscible, the binary plot, such as shown in Fig. 4, is useful. For later ease of calculation, it is desirable to express concentrations on a solute-free basis. If there is extensive miscibility, a ternary plot (Fig. 5) would be preferable. Tie lines represent the equilibrium between the coexisting phases. 

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